Contracts

Is it possible to have a legally binding contract without actually signing. For example, party a goes to web designer b and wants a page creating. All the communication is through the internet with no papers.

I mean there are ways to actually guard against people dropping out, for example, you accept half the payment before you start work and then the full payment before the work is transfered to them.

And in this "contract" what sort of things should be in it?

Yes. A verbal contract is one between two parties who exchange a verbal agreement to something. It is legally binding but I would think you would have to prove who agreed to what and that they agreed at all.
This is also acceptable but I don't understand, again, how you could prove the emails were not created by you if you saved them as text files. However, in court, keeping good records, even manually, is accepted in many cases as proof.
1) What you agree to do. 2) What he agrees to do. 3) For how much 4) By when? 5) What happens if you're late 6) When do you get paid 7) What is considered DONE. 8) How can either of you get out of the contract. 9) Who is supplying what?

10) Responsibility for implied costs

There's a slight chance that I could be getting burned by that. Protect yourself.